Fast decomposed method to devise broadband polarization-conversion metasurface

Designing a broadband, wide-angle, and high-efficient polarization converter with a simple geometry remains challenging. This work proposes a simple and computationally inexpensive method for devising broadband polarization conversion metasurfaces. We focus on a cross-shape configuration consisting of two bars of different lengths connected at the center. To design the metasurface, we decompose the system into two parts with two orthogonally polarized responses and calculate the response of each part separately. By selecting the parameters with a proper phase difference in the response between the two parts, we can determine the dimensions of the system. For designing broadband polarization conversion metasurfaces, we define a fitness function to optimize the bandwidth of the linear polarization conversion. Numerical results demonstrate that the proposed method can be used to design a metasurface that achieves a relative bandwidth of \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$97\%$$\end{document}97% for converting linearly polarized waves into cross-polarized waves. Additionally, the average polarization conversion ratio of the designed metasurface is greater than \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$91\%$$\end{document}91% over the frequency range of 10.9–28.5 GHz. This method significantly reduces the computational expense compared to the traditional method and can be easily extended to other complex structures and configurations.


Theory based on matrix operator
We apply matrix operators to analyze the linear optical response of the system. The Jones calculus is the mathematical formalism developed by R. C. Jones 1 . With this formalism, a Jones vector can represent the polarized incident light. A Jones matrix of the element can represent an optical element's linear optical effect on the incident polarization. Although Jones did not consider reflection from a surface or loss in the material in the original papers, we can use a similar mathematical formalism to analyze these cases.
As shown in Figure 1b, we will use two standard Cartesian coordinate systems in our analysis. The unit vector in the positive direction of each axis reads and The rotation matrix from the xOy coordinate to the uOv coordinate is written as where θ = 45 • is the angle between the two coordinate systems. In the uOv system, due to the spatial symmetry of the system, the reflection matrix can be written as corresponding to the reflection matrix in the xOy system where Φ u and Φ v are the polarized reflection phase for the uand v-polarized normal incidence, respectively. R uu and R vv are the reflection amplitude for the uand v-polarized normal incidence, respectively. The incidence is assumed to be linearly polarized in ydirection along the negative zdirection, which can be described as 1 To consider the effects of multiple surfaces, we can calculate the reflected wave using matrix multiplication as follows where R xy and R yy are the cross-and co-polarization reflection coefficients, respectively, when we have y-polarized incidence. The linear polarization conversion efficiency can be described using the polarization conversion ratio (PCR) as follows To simplify the analysis, we assume the system is lossless and there is no transmission, which means and Therefore, the PCR is determined by the phase difference between Φ u and Φ v , which can be achieved by adjusting the dimensions and arrangement of the metasurface elements.

Scattering from a linear rod antenna
We assume a thin linear antenna with a circular rod of radius a embedded in a homogeneous environment with refractive index n env = 1, made of a perfect electric conductor. We calculate the current distribution based on the boundary condition by solving Hallén integral equations for linear antennas. Then, we perform the integration to obtain the z-component of the magnetic vector potential, from which we determine the z-component electrical field. Here, we summarized the framework of our method. The external source field E in is assumed to be a uniform plane wave incident at an angle θ with respect to the z-axis on a receiving thin linear antenna. The z-component of the incident field is given by: where E 0 is the amplitude, θ is the incident angle, k is the wavenumber. We consider the case that the incidence impinges normal to the axis of the rod and polarizes along it. That is the θ = π/2, we get E in = E 0 . The current distribution on a thin linear antenna can be calculated using Hallén integral equation 2 . The boundary condition on the antenna surface (i.e., at ρ = a) requires that the z component of the total field vanish. We obtain Hallén integral equation as follows µ 4π

2/4
where ω denotes the angular frequency of the incidence, ε and µ denote the permittivity and permeability of the surrounding, respectively, k is the wavevector, G(z − z ′ ) = e −ikR R and R = a 2 + (z − z ′ ) 2 . Given the additional boundary condition at the ends of the antenna (I(z = l/2) = I(z = −l/2) = 0), we can obtain the current distribution I(z) using equation 14.
Once the current distribution is calculated, the z-component of the magnetic vector potential A z (z, ρ) can be determined using the following equation 2 Given the Lorentz gauge, the z-component of the electric field of a thin rod antenna can be readily calculated as follows We can approximate the second derivative in Equation 16 using the second-order central difference algorithm as follows: where h is the distance between neighboring x values on the discretized domain. We consider the vacuum case, where the relative permeability and permittivity are µ r = 1, ε r = 1. At a fixed incident wavelength, the phase shift between the emitted and the incident waves of a straight-thin antenna changes dramatically across a resonance. Figure S1b shows the phase and amplitude of the wave scattered from a straight rod antenna. However, it is concluded that the phase shift between the incident and scattered waves from a straight wire antenna changes within a range of π, which is insufficient for our purpose. One solution is to utilize the advantages of the V-shape antenna 3 . However, another issue arises in that wires with different lengths will have significantly different amplitudes. To address this, we implemented an alternative solution that combined straight wires with a PEC mirror to extend the phase difference beyond π and achieve approximately 99% reflection amplitude.